3.341 \(\int \frac{c+d x+e x^2}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=225 \[ -\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(x*(5*c + 4*d*x))/(18*a^2*(a + b*x^3)) - (a*e - b*x*(c + d*x))/(6*a*b*(a + b*x^3
)^2) - ((5*b^(1/3)*c + 2*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1
/3))])/(9*Sqrt[3]*a^(8/3)*b^(2/3)) + ((5*b^(1/3)*c - 2*a^(1/3)*d)*Log[a^(1/3) +
b^(1/3)*x])/(27*a^(8/3)*b^(2/3)) - ((5*b^(1/3)*c - 2*a^(1/3)*d)*Log[a^(2/3) - a^
(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(2/3))

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Rubi [A]  time = 0.415306, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} c-2 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x (5 c+4 d x)}{18 a^2 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2)/(a + b*x^3)^3,x]

[Out]

(x*(5*c + 4*d*x))/(18*a^2*(a + b*x^3)) - (a*e - b*x*(c + d*x))/(6*a*b*(a + b*x^3
)^2) - ((5*b^(1/3)*c + 2*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1
/3))])/(9*Sqrt[3]*a^(8/3)*b^(2/3)) + ((5*b^(1/3)*c - 2*a^(1/3)*d)*Log[a^(1/3) +
b^(1/3)*x])/(27*a^(8/3)*b^(2/3)) - ((5*b^(1/3)*c - 2*a^(1/3)*d)*Log[a^(2/3) - a^
(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(2/3))

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Rubi in Sympy [A]  time = 59.8174, size = 207, normalized size = 0.92 \[ - \frac{a e - b x \left (c + d x\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x \left (5 c + 4 d x\right )}{18 a^{2} \left (a + b x^{3}\right )} - \frac{\left (2 \sqrt [3]{a} d - 5 \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{2}{3}}} + \frac{\left (2 \sqrt [3]{a} d - 5 \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (2 \sqrt [3]{a} d + 5 \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d*x+c)/(b*x**3+a)**3,x)

[Out]

-(a*e - b*x*(c + d*x))/(6*a*b*(a + b*x**3)**2) + x*(5*c + 4*d*x)/(18*a**2*(a + b
*x**3)) - (2*a**(1/3)*d - 5*b**(1/3)*c)*log(a**(1/3) + b**(1/3)*x)/(27*a**(8/3)*
b**(2/3)) + (2*a**(1/3)*d - 5*b**(1/3)*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b
**(2/3)*x**2)/(54*a**(8/3)*b**(2/3)) - sqrt(3)*(2*a**(1/3)*d + 5*b**(1/3)*c)*ata
n(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(8/3)*b**(2/3))

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Mathematica [A]  time = 0.562098, size = 213, normalized size = 0.95 \[ \frac{\sqrt [3]{a} \sqrt [3]{b} \left (2 \sqrt [3]{a} d-5 \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (5 \sqrt [3]{a} \sqrt [3]{b} c-2 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{3 a \left (-3 a^2 e+a b x (8 c+7 d x)+b^2 x^4 (5 c+4 d x)\right )}{\left (a+b x^3\right )^2}-2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (2 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2)/(a + b*x^3)^3,x]

[Out]

((3*a*(-3*a^2*e + b^2*x^4*(5*c + 4*d*x) + a*b*x*(8*c + 7*d*x)))/(a + b*x^3)^2 -
2*Sqrt[3]*a^(1/3)*b^(1/3)*(5*b^(1/3)*c + 2*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/
a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(5*a^(1/3)*b^(1/3)*c - 2*a^(2/3)*d)*Log[a^(1/3) +
b^(1/3)*x] + a^(1/3)*b^(1/3)*(-5*b^(1/3)*c + 2*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2])/(54*a^3*b)

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Maple [A]  time = 0.006, size = 308, normalized size = 1.4 \[{\frac{cx}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,cx}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,c}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,c}{54\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,c\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,d{x}^{2}}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,d}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,d\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e{x}^{3}}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{e}{6\,ab \left ( b{x}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d*x+c)/(b*x^3+a)^3,x)

[Out]

1/6*c/a*x/(b*x^3+a)^2+5/18*c/a^2*x/(b*x^3+a)+5/27*c/a^2/b/(a/b)^(2/3)*ln(x+(a/b)
^(1/3))-5/54*c/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/27*c/a^2/b/
(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/6*d/a*x^2/(b*x^3+a
)^2+2/9*d/a^2*x^2/(b*x^3+a)-2/27*d/a^2/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/27*d/a^
2/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27*d/a^2*3^(1/2)/b/(a/b)^(1/
3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/6*e/a*x^3/(b*x^3+a)^2-1/6*e/a/b/(b*
x^3+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 4.37777, size = 163, normalized size = 0.72 \[ \operatorname{RootSum}{\left (19683 t^{3} a^{8} b^{2} + 810 t a^{3} b c d + 8 a d^{3} - 125 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{1458 t^{2} a^{6} b d + 675 t a^{3} b c^{2} + 40 a c d^{2}}{8 a d^{3} + 125 b c^{3}} \right )} \right )\right )} + \frac{- 3 a^{2} e + 8 a b c x + 7 a b d x^{2} + 5 b^{2} c x^{4} + 4 b^{2} d x^{5}}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d*x+c)/(b*x**3+a)**3,x)

[Out]

RootSum(19683*_t**3*a**8*b**2 + 810*_t*a**3*b*c*d + 8*a*d**3 - 125*b*c**3, Lambd
a(_t, _t*log(x + (1458*_t**2*a**6*b*d + 675*_t*a**3*b*c**2 + 40*a*c*d**2)/(8*a*d
**3 + 125*b*c**3)))) + (-3*a**2*e + 8*a*b*c*x + 7*a*b*d*x**2 + 5*b**2*c*x**4 + 4
*b**2*d*x**5)/(18*a**4*b + 36*a**3*b**2*x**3 + 18*a**2*b**3*x**6)

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GIAC/XCAS [A]  time = 0.220394, size = 301, normalized size = 1.34 \[ -\frac{{\left (2 \, d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{4 \, b^{2} d x^{5} + 5 \, b^{2} c x^{4} + 7 \, a b d x^{2} + 8 \, a b c x - 3 \, a^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

-1/27*(2*d*(-a/b)^(1/3) + 5*c)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^3 + 1/27
*sqrt(3)*(5*(-a*b^2)^(1/3)*b*c - 2*(-a*b^2)^(2/3)*d)*arctan(1/3*sqrt(3)*(2*x + (
-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^2) + 1/18*(4*b^2*d*x^5 + 5*b^2*c*x^4 + 7*a*b*d
*x^2 + 8*a*b*c*x - 3*a^2*e)/((b*x^3 + a)^2*a^2*b) + 1/54*(5*(-a*b^2)^(1/3)*a*b^3
*c + 2*(-a*b^2)^(2/3)*a*b^2*d)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^4)